Binary Calculator With Overflow


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Floating Point Numbers Computerphile

People expect computers to be entirely accurate and precise with numbers. You know, you grow up with calculators.You expect that if you type in: quot;0.1 + 0.2quot;, you expect it to come out to quot;0.3quot;. And sooner or later every programmer, as they're learning, tries something like that. And they type it into a programming language. Something like quot;0.1 + 0.2quot;,and they will hit Enter, and what the computer will give themis something like quot;0.300000000.1quot;. And at that point, they will have started to learn about floating point numbers,

and they will be starting to tear their hair out. Floating point numbers are essentially scientific notation. For the uninitiated, let's sayyou're an astrophysicist, and you have the speed of light, which is this enormous number with eight 0s after it. And you want to multiply it by a tiny distance. It doesn't really matter what we're measuring here, it's just a big number and a small number. Let's say, 0.00000015.

Let's say you want to multiply these two together.(I may have made this a bit deliberately easy.) You could try and do some really complicated maths and work it all out, but you don't need to.Because in scientific notation, this is 3 times 10 to the 8, and this one is 1.5 times 10 to the 7. So you don't need to do anything particularly complicated to multiply these two together. All you have to do is multiply these,which is 4.5, and then add these exponents together here.

So, 8 add 7, eight minus seven means that's 1. So we'll have 4.5 times 10 to the 1, or 4.5 times 10, or 45. Brilliant. That's much easier than trying to actually handle large numbers. And it also brings us to the concept of significant figures. Because the speed of light isn'texactly 3 times 10 to the 8. It's 2.997something, and there's a lot of digits on the end of there which don't really matter for everyday calculations. We round it up. We say it's 3 times 10 to the 8.

Or maybe, for being precise, we say it's 2.997 times 10 to the 8, but we don't really worry about the numbers beyond that. Hold that thought. So, we'll come back to significant digits later. The two main advantages to floating point are speed and efficiency. Speed, because floating point has been built over many, many years. It is now lightning fast for computers to deal with. And efficiency, because it can deal with really big numbers massive numbers, quot;size of the universequot; numbers

and really small numbers quot;size of an atomquot; numbers without needing enormous amounts of space. If you had a format that needed to include the size of the universe and the size of an atom, you'd need so many zeroes stored on both ends or you'd need something particularly complicated for a computer to work out that it would either be really inefficient, storing one number as a megabyte, or you would need something that would bereally fiddly for a computer to work out as it tries to place all the zeroes and the numbers everywhere.

How to convert from Binary to Hexadecimal

In previous tutorial, we discussed aboutbinary to octal conversion, in this tutorial we will discuss about binary tohexadecimal conversion. For binary number base or radix is 2 and it containsonly two numbers that is 0 and 1. For hexadecimal number base or radix is16 and it contains only 16 numbers. Those are 0 1 2 3 4 5 6 7 8 9 and alphabet A B C DE and F. Now here 10 is represented as A, similarly 11 as B, 12 as C, 13 as D, 14 as Eand 15 as F. Hexadecimal numbers can be represented as 4 bit. For first bitis 010101010101010 and 1, for second bit is 2 times 0, 2 times 1 again 2 times 0, 2 times 1, again 2 times 0, 2 times 1, again 2

times 0, 2 times 1, for third bit it is 4times 0, 4 times 1, again 4times 0, 4 times 1, for 4th bit it is 8 times 0 and 8 times1. So 0 can be represented as 0000, 1 can be represented as 0001, 2 can be represented as 0010, 3 can berepresented as 0011, 4 can be represented as 0100, 5 can berepresented as 0101, 6 can be represented as 0110, 7 can be represented as 0111, 8 can be represented as 1000, 9 can be represented as 1001, Acan be represented as 1010, B can be represented as 1011, C can berepresented as 1100, D can be represented

as 1101, E can be represented as 1110and F can be represented as 1111. Now let's convert Binary 1110 1 into Hexadecimal number. For binary number 111 01, the leftmost 1 is MSB and therightmost 1 is LSB. Now do the 4 digit grouping from LSB to MSB. So this 1this is 0 this one and this one comes into a group and this 1 makes anothergroup, but only 1 digit available here. So add three zeros to the lefthand sideto complete the fourdigit grouping. Now this is 0001 is 1 and this 1101 is13 that is D. Here this 1 is MSB and this D is LSB. So after arranging from MSB to LSB we get Hexadecimal number 1D. So for

Binary to Hex conversion Binary 11101 is equal to Hexadecimal1D. Now verify the result with calculator. For that click on Start menu type calchere and press Enter button on the keyboard, then calculator window willcome click on view then click programmer nowselect Bin radio button type binary 11 101 now select Hex radio button and theanswer shows here is 1D. So our Binary to Hexadecimal conversion is correct. Homework for you. Convert Binary 1011001 into Hexadecimal number and share it withthe comment section given below. In the

next tutorial I am going to discuss aboutOctal to Hexadecimal and Hexadecimal to Octal conversion.

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